raster convergence with respect to a closure...
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CAHIERS DETOPOLOGIE ET GÉOMÉTRIE DIFFÉRENTIELLE
CATÉGORIQUES
E. GIULI
J. SLAPALRaster convergence with respect to a closure operatorCahiers de topologie et géométrie différentielle catégoriques, tome46, no 4 (2005), p. 275-300<http://www.numdam.org/item?id=CTGDC_2005__46_4_275_0>
© Andrée C. Ehresmann et les auteurs, 2005, tous droits réservés.
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RASTER CONVERGENCE WITH RESPECT TOA CLOSURE OPERATOR
by E. GIULI and J. SLAPAL
CAHIERS DE TOPOLOGIE ETGEOMETRIE DIFFERENTIELLE CATEGORIQUES
Volume XL VI-4 (2005)
Resume. Nous introduisons et 6tudions le concept de convergencesur une cat6gorie concrete IC par rapport a un op6rateur c de cl6turesur IC. Nous commengons en définissant et examinant les voisi-
nages des sous-objets d’un K-objet donn6 par rapport a c. En suite,les voisinages sont utilis6s pour 1’ introduction de la convergence à1’ aide de certains filtres généralisés. Quelques proprietes de basesont en suite discut6es et la separation et compacit6 sont 6tudi6esen plus detail. Nous montrons que la separation et compacit6 in-duites par la convergence se comportent d’une facon analogue a laseparation et compacit6 des espaces topologiques et qa d’une fagonplus d6cente que la c-s6paration et la c-compacit6 usuelle.
0 Introduction
In this paper, we continue the study of closure operators on categories in thesense of D. Dikranjan and E. Giuli [12]. At present, the theory of closureoperators on categories is an important branch of categorical topology anda number of authors have contributed to its development. Categories withclosure operators generalize the category Top and it was shown by some ofthese authors that many topological concepts can be naturally extended fromtopological spaces to objects of categories with closure operators. For exam-ple, separation and compactness are extended and studied in [9],[10],[13],connectedness in [3], [4] and [7], quotient maps in [11] and openness in [17].But, up to now, no attempt has been made to define and study a convergence
The second author gratefully acknowledges financial supports from NATO-CNR (Out-reach Fellowship No. 219.33), from Grant Agency of the Czech Republic (Project No.201/03/0933) and from the Ministry of Education of the Czech Republic (Project No.MSM0021630518).
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with respect to a closure operator on a category. And this is what we areconcerned with in this paper.
Categories equipped with certain types of convergence are dealt within [19-23]. In particular, in [22], a general concept of convergence on acategory is introduced and discussed and it is shown that the convergenceinduces, under some natural conditions, a closure operator on the category.The aim of the present paper is just converse - we will show that a closureoperator on a category gives rise to a convergence on the category. This
convergence will then be investigated. But, instead of using the approach of[19-23], which is based on employing (generalized) nets for expressing theconvergence, we rather use the more common approach based on employing(generalized) filters.
Given a concrete category K with a closure operator c, we start with in-
troducing the notion of a neighborhood of a subobject of a K-object. Theseneighborhoods, which generalize the usual neighborhoods in Top, are notclosed under finite meets in general. So, when we use them for defining aconvergence with respect to c on (objects of) )C, we have to employ certainstructures that are more general than filters. The structures employed arecalled rasters and the defined convergence is referred to as the raster con-
vergence (with respect to c). The raster convergence is investigated and itis shown that it behaves analogously to the filter convergence in topologi-cal spaces. Focus is then put on the study of raster separation and rastercompactness with respect to c, i.e., separation and compactness induced bythe raster convergence with respect to c. Results analogous to some classicalones on separation and compactness in topological spaces are proved includ-ing the "minimality", "absolute closedness" and "Tychonoff’s theorem". Itfollows that raster separation and raster compactness with respect to c behavebetter that the usual c-separation and c-compactness. Relations between thetwo kinds of separation (respectively, compactness) are also discussed in thepaper.
1 Preliminaries
For the basic categorical terminology used see [1] and [14].Throughout the paper, x denotes a finitely complete category with a
proper (E, M) -factorization structure for morphisms.
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The condition that (E, M) is proper means that £ is a class of X-epimor-phisms and Jvl is a class of X-monomorphisms (which then contains allextremal X-monomorphisms). It is imposed only for the reason of makingformulations of the presented results as brief as possible. For instance, thecondition implies that M contains all points of any X-object X, i.e., X-morphisms 1X-> X (by lx we denote an arbitrary but fixed terminal objectof X). But it also results in each of the following two conditions: (1) forevery X-object X the diagonal morphism dx = idX, idx >: X--> X x X
(where idx : X -> X denotes the identity morphism) belongs to At (2)g o f E M => f E M. It will be clear from the context which results remainvalid even when (E, M) is not proper, i.e., when it is supposed only that Mis a class of X-monomorphisms.
Note that M is closed under products and pullbacks (along arbitrary X-morphisms). We assume that X has multiple pullbacks of arbitrary largefamilies of M-morphisms with a common codomain. In this case, M isclosed under multiple pullbacks. Given an X-object X, each M-morphismwith codomain X is called a subobject of X . We denote by subX the (pos-sibly large) complete lattice of all isomorphism classes of subobjects of X.As usual, we identify isomorphism classes of subobjects of X with theirrepresentatives. So, each subobject of X is considered to be an element ofsubX, and we write m = n instead of m = n for subobjects m, n of X.In the same way, by saying that m and n are different, in symbols m # n,we mean that m and n are not isomorphic. The joins and meets in subXwill be denoted by the usual symbols V, V and A, 1B, respectively. The leastelement of subX is denoted by ox. If idx = ox, then the X-object X iscalled trivial. Of course, if .6 is stable under pullbacks, then an X-object Xis non-trivial if and only if there exists an E-morphism f : X -> Y with Y
non-trivial (because then m1 # m2 implies f-1(m1) = f-1 (m2) wheneverml, m2 E subY).
We suppose that, for each X-object X, the lattice subX is pseudocom-plemented. This means that for each m E subX there exists m E subX suchthat the equivalence m A n = ox=> n m is valid whenever n E subX.The (unique) morphism m is called the pseudocomplement of m. Clearly,given m, p E subX, we have m p => p m and m m. It also followsthat subX is a (possibly large) Boolean algebra (with complements given bypseudocomplements) if and only if m = rn for all m E subX (cf. [18]) .
Note that, since subX is pseudocomplemented, any atom p E subX has
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the property p m or p m whenever m E subX (because p 1:. m =>
p A m = ox =>pm).In the category X, each morphism f : X- Y is assumed to fulfill the
following two axioms:
, whenever m E subX,
I whenever n E subY.
Of course, 1° is equivalent to f-1 (oY) = oX.We will need the following observations:
Lemma 1.1 Let f : X-> Y be an X-morphism, m E subX and p E subY.If oY p f (m), then ox m A f-1 (P).
Proof. Let oy p f (m) and suppose that m A f-1 (p) = ox. Thenm f-1 (P), hence , Consequently,
and thus p A f(m) = oL. But this is a contradiction with
Lemma 1.2 Let f : X-> Y be an X-morphism and m E subX be an atom.Then f (m) E subY is an atom too.
Proof. Let p E subY, 0 p f (m). By Lemma 1.1, ox m A
f-1(p). Thus m A f -1 (p) = m which yields m f -1 (p). Hence f(m)f(f-1(p)) p. Consequently, f (rn) = p. Therefore f (m) E subY is anatom. D
Lemma 1.3 Let X = I1iEI Xi be a product in X and pi E subxi be an atomfor each i E I. If all pi, i E I, have the same domain (up to isomorphisms),then (pi; i E I) E subX is an atom too.
Proof. Let Z denote the domain of.all pi, i E I. As Z is non-trivial, thereholds (pi; i E I) > ox. Let m E subX, ox m (pi; i E I). Then thereexists r E subZ, r > oz, with m = (pi; i E I)or. Hence pi or = pi(r) > oxfor each i E I. Thus, since pi o r pi, we get pi o r = pi for each i E I.Consequently, m = (pi; i E I)or = (pior; i E I) = (pi; i E I). Therefore,(pi ; i E I) E subX is an atom. D
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Remark 1.4 Let X be an X-object. If card ( sublx) = 2, then any point x ofX is an atom of subX. Conversely, if every non-terminal X-object Z has theproperty card(subZ) # 2, then any atom of subX is a point of X. Thus, forany X-object X, the points of X coincide with the atoms of subX providedthat terminal objects of X are just the X-objects T with card(subT) = 2.
Further, we suppose there is given a concrete category 1C over X withthe corresponding underlying functor | | : K-> X . As usual, we donot distinguish notationally between 1C-morphisms and their underlying X-morphisms (i.e., we write f instead of |f| whenever f is a 1C-morphism).Given a 1C-object K, by a subobject of K we will always mean a subobjectof |K| and correspondingly, we will write briefly subK and oK instead ofsubiki and o|K|, respectively. This will cause no confusion because onlythe category X, and not /Ç, is assumed to have a subobject structure. Thecategory )C is also supposed to have finite concrete products, and by a (notnecessarily finite) product in JC we always mean a concrete one. The classof all ,M-embeddings in 1C, which are called briefly embeddings, is denotedby Embm.
Recall that a closure operator on 1C (with respect to (E, M)) is a familyof maps c = (CK : subK - subK)KEK with the following properties thathold for each IC-object K and each m, p E subK:
(3) f(cK(m)) cL(f(m)) for each K-morphism f : K-> L.
Note that, for any X-morphism f : K - L and any m E subK, f (m) isthe M -part of the (6, M) -factorization of f o m. In fact, the above definedclosure operator c on K is a so-called U-closure operator [2] where U = 1 I.A classical closure operator introduced in [12] is obtained when k = X and
| | is the identity functor.The closure operator c is called
(a) grounded if CK (OK) = OK for each K E K:,
(b) idempotent if cK(cK(m)) = cK (m) for each K E IC and each m EsubK,
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(c) additive if cK (m V p) = cK (m) V cK (p) for each K E 1C and eachm, p E subK,
(d) hereditary if, whenever m : M - K is an embedding in IC, cm (p) =m-1 (cK(m o p)) for each p E subM.
Given a IC-object K, a subobject m E subK is said to be c-closed (respec-tively, c-dense) provided that cK(m) = m (respectively, cK(m) = idK). AIC-mozphism f : K - L is called c-preserving if f (cK(m)) = cL( f (m))whenever m E subK. Thus, if f is c-preserving, then it maps c-closed sub-objects to c-closed subobjects, and vice versa provided that c is idempotent.
Throughout the paper, we assume there is given a grounded closure oper-ator c = (cK)KEK on IC. Given a pair K, L of K-objects with I K I = |L|, weput cK CL provided that cK(m) cL(m) for each m E subK(= subL).
Example 1.5 (1) Basic examples of the above introduced category IC witha closure operator are certain topological constructs with X = Set whereI I : )C -> Set is the forgetful functor and the (surjections, injections)-factorization structure for morphisms is considered in the base category Set.A number of such examples are given in [2], [10], [13], [14]. Among them,of course, the most natural one is K = Top, i.e., the construct of topo-logical spaces and continuous maps, with the Kuratowski closure operator.Further examples can be found among concrete categories over topologi-cal constructs (with a singleton fibre of the empty set) which always havethe (surjections,embeddings)-factorization structure for morphisms. For in-stance, let X = Top, let K be the category TopGrp of topological groups(and continuous homomorphisms), and let || : TopGrp -> Top be theforgetful functor (that forgets the group structure). A closure operator onTopGrp is given by the (classical) Kuratowski closure operator on Top.On the other hand, TopGrp equipped with the classical Kuratowski closureoperator (i.e., the closure operator where | | : TopGrp -> TopGrp is theidentity functor) is not an example of 1C because TopGrp does not fulfillthe axiom 1°.
(2) Recall that a projection space is a pair (X, (an)nEN) where X is a set and(an)nEN = (an : X -> X)nEN is a sequence of maps such that an o am =amin(m,n). Given projection spaces (X, (an)nEN) and (Y, (Bn)nEN), a mapg : X -> Y is called a projection function of (X, (an)nEN) into (Y, (Bn)nEN)provided that Bn o g = g o an for all n E N. Projection spaces (X, (an)nEN)
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with an = f for all n E N, where f : X -+ X is a map, coincide with idem-potent mono-unary algebras. Let JC be the category of projection spaces andprojection functions.(a) Let X = IC, let : K-> K be the identity functor, and consider the(surjections,injections)-factorization structure for morphisms in IC. With re-spect to this factorization structure, there is a closure operator c = (CK )KEICon IC given by cK(m) = {x E X;dn E N : an (x) E m(M)} wheneverK = (X, (an)nEN) is a projection space and m : M - K is a subobjectof K. This closure operator coincides with the closure operator Coo from
[16]. So, by [16], c is idempotent, additive and hereditary. It can easily beseen that, given a IC-object K, subK need not be a Boolean algebra (e.g.,let K = (X, f ) be the idempotent mono-unary algebra with X = {0,1},f (0) = 1 and f (1) = 1).(b) Let X = Set, let || : k -> X be the forgetful functor and consider the(surjections,injections)-factorization structure for morphisms in Set. Withrespect to this factorization structure, there is a closure operator c = (CK)KEKon /C given by cK(m) = m(M)U{x E X; BIn EN : an(x) E m(M)} when-ever K = (X, (an)nEN) is a projection space and m : M -3 X is a subobjectof K. Moreover, c is clearly idempotent and hereditary. It is a so-called non-standard closure operator - see [2]. Of course, for those subobjects m of Kwhich coincide with (underlying sets of) subobjects of K in the sense of (a),cK (m) coincides with (the underlying set of) cK (m) from (a).(c) Let the situation be the same as in (b). Then, with respect to the factoriza-tion structure considered, there is another closure operator c = (CK )KEIC onIC defined as follows: cK(m) = m(M) U {an (x) ; x E m(M) and n E N}whenever K = (X, (an)nEN) is a projection space and m : M-> X is a
subobject of K. Clearly, this closure operator is not only idempotent andhereditary, but also additive. Therefore, it is more appropriate than the clo-sure operator c from (b). It is also obvious that CK-closed subobjects of aIC-object K (i.e., subsets of K) coincide with the subobjects of K from (a).In other words, c is a so-called hull operator - see [2].
In what follows, whenever a construct is taken as an example of IC with-out mentioning the functor II : IC -> X, we always suppose that X = Set,II : : )C -3 Set is the forgetful functor, and the (e, M) -factorization structurefor morphisms in the base category Set is just the (surjections,injections)-factorization structure. In the case K = Top we moreover suppose that c isthe Kuratowski closure operator.
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Remark 1.6 Example 1.5(2) demonstrates one of the advantages of U-clo-sure operators over the classical closure operators - the former often act onricher domains than the latter (cf. also Example 2.3(2)). Another advantageis that, when studying U-closure operators where U is an underlying functor,we can use concepts like an embedding, a fibre, etc. to describe behavior ofthese operators (cf. 4.10-4.12).
2 Neighborhoods and rastersDefinition 2.1 Let K be a 1C-object and m E subK. A cK-neighborhood ofm is any subobject n E subK such that m cK(n).
We denote by ACK (M) the class of all cK-neighborhoods of m. If it isclear which lC-object K is considered, we will call cK-neighborhoods brieflyneighborhoods and instead of NCK(m), will write briefly N(m).
Proposition 2.2 Let K be a K-object and m, p E subK. Then
(4) n E N(m) implies m n provided that m is an atom of subK orn=n,
(9) if nl, n2 E JV(m), then n, A n2 E N(rn) provided that c is additiveand subK is a Boolean algebra.
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Proof. Let n E N(m) and m A n = OK. Then m iff cK(n). Thus, sincem cK(n), we have m cK(n)Ack(n) = OK. Hence (7) is valid for k = 1.Suppose that it is valid for some k E N and let nl, n2, ..., nk, nk+1 E N(m).Then m A n1 A n2 A ... A nk > OK and nk+1 E N(m A n, A n2 A ... A nk),thus m A nl A n2 A ... Ank A nk+1 >Ok (because (7) is valid for k = 1). This
proves (7), and (8) is an immediate consequence of (7). The other conditionsare obvious. 0
Example 2.3 (1) Of course, if K = Top, the above defined neighborhoodscoincide with the usual neighborhoods (of sets).(2) In Example 1.5(2), let K E K be the projection space K = (N, (an)nEN)where, for each n, p E N, an(p) =min(n, p).If c is the closure operator on K given in the part (a) of the Example, thenn > oK=> n E N(m) whenever m, n are subobjects of K (because thesubobjects of K are c-closed and coincide with the subsets of N having theform {x E N ; x n} where n E N U oo, so that n > oK=> n = Ok for
each subobject n of K).On the other hand, if c is the closure operator on K given in the part (b),then CK coincides with the discrete topology on N. Therefore, we have n EN(m) => m n whenever m, n are subobjects of K.Finally, let c be the closure operator on JC from the part (c) of Example1.5(2). Let m : M - N be an arbitrary subobject of K with m > oK.
Then one can easily see that CK(m) = N if m(M) is infinite, and CK(M) =( 1 , 2 , ... , max m(M)} if m(M) is finite. Consequently, we have N(m)={N C N ; x E N for each x E N with x> min m(M)}. Thus, if x E N is apoint, then {y E N; y> x} is the smallest neighborhood of x. It follows that(N, CK) is nothing but the so-called right topology on (the linearly orderedset) N.
Proposition 2.4 Let f : K - L be a JC-morphism, m E subK and n EN( f (m) ). Then f-1 (n) E N(m).
Proof. Assume f-1 (n)E N(m). Then and therefore,
which yields f(m) cL(n). But this is a contradiction with
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Proposition 2.5 Let K be a )C-object and m, p E subK, m > OK. If m CK (P), then n A p > OK for each n E N(rn), and vice versa provided thatsubK is a Boolean algebra and m is an atom of subK.
Proof. Let m cK(p) and admit that there exists n E N(m) with n A p =oK. Then p n, hence cK(p) cK(n). Thus, since m A cK(n) = OK, wehave m A cK (p) = oK. But this is a contradiction with oK m cK (p).Vice versa, let n A p > oK for each n E N(m) and let m be an atom ofsubK. Admit that m cK(p) = cK(p). Then rn A CK (P) > OK, which
yields m A cK(p) = m. Hence m cK(p) = cK(p), thus 15 E N(m). Butp A P = oK, which is a contradiction. Therefore m CK(P). D
Corollary 2.6 Let K be a )C-object such that subK is a Boolean algebraand let p E subK. IfcK(p) equals a join of atoms of subK, then cK(p)V{m E subK; m is an atom and n A p > OK for each n E N(m)}.
Corollary 2.7 Let K, L be 1C-objects with IKI = ILl and let subK (= subL)be an atomic Boolean algebra. If NCL (m) c NCK (m) for each m E subK,then cK cL.
Remark 2.8 a) In Proposition 2.5 and Corollary 2.6, the condition that subKis a Boolean algebra can be replaced by the weaker condition that p = p andCK (P) = CK (P).b) Having defined cK-neighborhoods (K E )C an object), we can define c-open subobjects of K as those m E subK that fulfill m E NcK (m) or equiv-alently, m E NcK (p) whenever p E subK and p m. Then, obviously, anarbitrary element m E subK is c-open if and only if m is c-closed. It is alsoevident that the inverse image of a c-open subobject under a 1C-morphismis c-open (in consequence of the conditions 1° and 2° in the previous sec-tion). But c-open subobjects have already been introduced in [17] as thosem E subK that satisfy m A cK(p) cK(m A p) whenever p E subK.This concept of c-openness is stronger than the above defined one (as c isgrounded) and both concepts coincide provided that c is additive and subKis a Boolean algebra. In this note, we do not need any concept of c-opennessand so we will avoid using it.
Definition 2.9 Let 9 be a (possibly large) complete lattice with the leastelement 0. A subclass R C 9 is called a raster on g provided that
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(1) R is an upper class of g (i.e., x E R implies y E R for every y E g,y > x), and
(2) R is centered (i.e., R # 0 and every finite meet of elements of R isdifferent from 0).
Given a (possibly large) complete lattice!g, we will use the usual con-cepts of filter, ultrafilter, filter base, ultrafilter base and filter subbase natu-rally extended to subclasses of G. Thus, every filter on 9 is a raster on g.Conversely, every raster on 9 which is a filter base on g is a filter on g. Ofcourse, filter subbases on G coincide with centered subclasses of g. If R, Sare rasters on G, then S is said to be finer than R, and R is said to be coarserthan S, provided that R C S.
For any subclass B C 9 we put [B] = {y E g; 3x E B : x y} andB= {x ; x is a finite meet of elements of B}. Thus, if B is centered, then [B]is a raster on g. If B is a raster on g, then 8 is a filter base on g, so that [8]is a filter on G. Therefore, for each raster R on g there is a filter on g whichis finer than R. It follows that maximal (with respect to "coarser") rasters ong coincide with ultrafilters on G. As the Axiom of Choice for conglomeratesis supposed, each raster on 9 is coarser than an ultrafilter on g.
For each X-object X we denote by Rx the conglomerate of all rasters onsubX. Thus, Rx = 0 if and only if X is a trivial object (because otherwise{idX} E RX). Given a 1C-object K, we write briefly RK instead of RIKI.
Let X, Y be X-objects and B C subX a subclass. As usual, if f : X-Y is an X-morphism, we put f (B) = {f (r) ; r E B}. We then clearly have[f [B]] = [/(8)]. If B is a centered subclass or a filter base or an ultrafilterbase respectively, then so is f(B) (for ultrafilter bases this follows from thefact that a centered class R C subX is an ultrafilter if and only if m E R orm E R for each m E subX). If 6 is stable under pullbacks and f E 6, thenf(B) E Ry whenever B E RX.
Let X = lliEI Xi be a product in X and let Bi C subX, for each i E I.Then we put IIiEI Bi = trIiEI mi; mi E Bi for each i E I}. If in X thenon-trivial objects are stable under products and if Bi is centered for eachi E I, then HIEI Bi E subX is centered too (becauseAjEJ fliEI mi >
IIiE1I A jEJ mji whenever 7YPi E Bi for each j E J and each i E I, and thedomain of AJEJ mji is non-trivial for each i E I), hence [IIiEI Bi] E RX.
The following, quite obvious fact will be used later on.
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Remark 2.10 If each non-trivial X-object has a point and card(sub1x) > 2,then in X the non-trivial objects are stable under products.
Let K be a JC-object and m E subK, m > oK. By Proposition 2.2,N(m) is a raster on subK (and N(m) is even a filter provided that c isadditive and subK is a Boolean algebra). But N(m) need not be a filteron subK in general. For this reason the notion of a raster is introduced andconvergence is defined further on in terms of superclasses of N (m).
3 Raster convergence
Definition 3.1 Let K be a IC-object, m E subK and R E RK.
(a) We say that R converges to m, and write R-> zn, if N(P) C R foreach p E subK with oK p m.
(b) m is called a clustering of 1(, provided that m cK (r) for each r E R.
Clearly, there holds:
Proposition 3.2 Let K be a )C-object and m E subK. Then
(1 ) R-> oK for each R E RK.
(2) N(m) --t m whenever m is an atom of subK.
(3) For any R E RK and any m E subK, from R -4 m it follows thatR -> p for each p E subK, p m.
(4) Let the lattice subK be atomic, let R E RK and let m E subK. IfR-> a for each atom a E subK with a m, then 1(, --t m.
(5) For any R E RK and any m E subK, from R-> m it follows thatS-> m whenever S E RK is finer than R.
(6) OK is a clustering of every raster R E RK.
(7) Let R E RK and m, n E subK. If m is a clustering of R and n m,then n is a clustering of R too.
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Example 3.3 Let IC = Top, let K be a IC-object, R E RK be a filter, andm : M - IKI be an inclusion (in Set). Then R-> m (respectively, m is aclustering of R) if and only if R converges to x (respectively, x is a clusterpoint of R) - in the usual topological sense - for each x E M.
As an immediate consequence of Proposition 2.5, Definition 3.1 andProposition 3.2 we get
Proposition 3.4 Let K be a IC-object, rn, p E subK, and let m be an atomof subK. If m CK(P), then there exists R E RK such that R - m andn A p > OK for each n E R, and vice versa provided that subK is a Booleanalgebra.
Corollary 3.5 Let K be a IC-object such that subK is a Boolean algebraand let p E subK. If cK(p) equals a join of atoms of subK, then CK (P) =Vfm E subK; m is an atom such that there exists R E RK with R-> mand n A p > oK for each n E R}.
Proposition 3.6 Let K be a IC-object such that subK is a Boolean algebra,let R E RK and let rn E subK be a join of atoms. If there exists S E RKwith R C S and S - m, then m is a clustering of R, and vice versaprovided that subK is atomic, c is additive and R is a filter.
Proof. For m = OK the statement is trivial. Let m > OK and let thereexist S E RK with R C S and S - m. Then, for an arbitrary atomp E subK with p m, we have S-> p. From N(p) C S it follows thatr A n > OK whenever r E R and n E N(p). By Proposition 2.5, p cK(r)for each r E R. Hence p is a clustering of R, i.e., p A{cK(r) ; r E R}.Consequently, m is a clustering of R.Conversely, let subK be atomic, c be additive and R be a filter. Suppose thatm is a clustering of R and let p m be an arbitrary atom of subK. Put8 = {r A n; r E R, n E N(p)}. By Proposition 2.5, r A n > OK wheneverr E R and n E N(p). As R is a filter and by Proposition 2.2(9), N(p) isa filter too, 8 is a filter base. We have R C [8] and N(p) g [,6], hence[B] - p. By Proposition 3.2(4), [B] -> m. The proof is complete. 0
Corollary 3.7 Let K E JC be an object such that subK is a Boolean algebra,let R E RK and let m E subK be a join of atoms. If R -> m, then m is aclustering of R.
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Corollary 3.8 Let c be additive, K be a IC-object such that subK is anatomic Boolean algebra, and let R E RK be an ultrafilter. Then R - m ifand only if m is a clustering of R.
In the last section, we will need the following result.
Lemma 3.9 Let c be additive, K be a IC-object such that subK is a Booleanalgebra, and let m, p E subK where m is an atom. Then the followingconditions are equivalent:
(1) m CK(P).
(2) There exists a filter base B C subK with [B] -> m such that q P foreach q E B.
(3) There exists a filter F E RK such that F -> m and p E F.
Proof. Let (1) be true. Then, by Proposition 2.5, n A p > OK for eachn E N(m). By Proposition 2.2(9), N(m) E RK is a filter. Thus, B =
{n A p; n E N(m)} is a filter base and q p for each q E Li. Since
N(m)-> m (by Proposition 3.2(2)) and N(m) c [B], we have [B] - m.Thus, (1)=>(2).The implication (2)=>(3) is obvious.Assume that (3) is true. Then n A p > OK for each n E T and by Proposition3.4, we get m CK(p). Hence (3)=>(1). ~Theorem 3.10 Let f : K -> L be a IC-morphism, m E subK and R E RK.If R -> m, then [f (R)] -> f (m).
Proof. Let R -> rn, p E subL, oL p f (m), and let n E N(p). Sincef(f-1(P)) p, we have n E N(f(f-1(p))). Thus, by Proposition 2.4,f-1(n) E N(f-1(p)). Since f-1(p) A m f-1(p), we have f-1(n) EN(f-1(p) A m). By Lemma 1. l, OK f-1(p)a m m. Thus, .N( f -1 (p) nm) C R because R - m. It follows that f -1 (n) E R, hence f(f-1(n)) Ef(R) C [f(R)]. As n > f(f-1(n)), there holds n E [fCR)]. Therefore,N(p) ç [f(R)], which yields [f(R)]-> f(m).~
Let K = IliEI Ki be a product in )C and let R E RK. By Theorem 3.10,given m E subK, R - m implies [pri(R)] - pri (m) for each i E I. If theconverse implication is also valid, we say that the raster R is convergence-compatible with the product K. The following statement provides a usefulcriterion for the convergence-compatibility of filters:
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Proposition 3.11 Let K = DiEI Ki be a product in IC. For any p E subK,OK p, and any n E N(p), let there exist a finite subset Io g I and asubobject ni E subki with ni E N(ni) for each i E Io such that p
AiEIopri-1(ni) n. Then every filter on sub K is convergence-compatiblewith K.
Proof. Let R E RK be a filter, m E subK and (pri(’R)] J -> pri(m) for eachi E I. Suppose also that p E subK, OK p m, and let n E N(p). Thenthere is a finite subset Io C I and a subobject ni E subKi with ni E N(ni)for each i E Io such that p :5 AiEIo pri- 1(ni) n. Consequently, pri(p) AjEIo priprj-1 (nj) pripri-1(ni) :5 ni for each i E Io. It follows that, foreach i E Io, ni E N(pri(p)) because ni E N(ni). Since oK, pri(p) pri(m), we have ni E (pri(’R)] for each i E Io. Thus, whenever i E Io,ni > pri(r) for some r E IZ. Hence pri-1(ni) > pri-1 pri(r) > r, so that
pri-1(ni) E IZ for each i E Io. Therefore, AiEIo pr-1(ni) E IZ, which yieldsn E IZ. It follows that N(p) C R. We have shown that R-> m. Thisproves the statement. 0
Example 3.12 For IC = Top, the assumptions of Proposition 3.11 are clearlyfulfilled. Thus, we get the well-known fact that filters are convergence-compatible with products of topological spaces.
Proposition 3.13 Let in X the non-trivial objects be stable under productsand let K = DiEI Ki be a product in K. For each i E I, let ’Ri E RKi,mi E subKi and Ri--> mi. If [DiE I Ri] E RK is convergence-compatiblewith K, then [
Proof. By the assumptions, [IIiEI Ri] E RK. Let ri E 7Zi, ri : Ri-> Ki foreach i E I, and put R = IIiEI Ri. Then, for each i E I, pri o IIiEI ri = ri o piwhere pi : R - Ri is the projection. Thus, pri(TIiEI ri) is the M-part of the(E, M) -factorization of ri o pi. Now, the diagonalization property results inPri (rIiE, ri) ri for each i E I. Analogously, we have pri (IIiEI mi) mifor each i E I . Hence [pri(IIiEI Ri)] 2 Ri, which yields (pri(TIiEI Ri)] -mi for each i E I. Consequently, ! )) foreach i E I. Since [IIiEI Ri] is convergence-compatible with K, we have
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4 Raster separation and raster compactnessDefinition 4.1 A K-object K is said to be
(a) raster separated (with respect to c) provided that, whenever m, p EsubK are atoms and R E RK, from R -+ m and R-> p it followsthat m = p,
(b) raster compact (with respect to c) provided that each 7Z E RK has aclustering different from oK (or equivalently, has the propertyA{CK(r) ; r E R} > OK).
Example 4.2 In the case K = Top, a topological space is raster-separated(respectively, raster compact) if and only if it is a Hausdorff space (respec-tively, compact in the usual sense).
Theorem 4.3 Let K be a IC-object. If for any pair p, q E subK of differentatoms there exist m E N(p) and n E N(q) such that m A n = OK, then Kis raster separated, and vice versa provided that c is additive and subK is aBoolean algebra.
Proof. Assume that, for any pair p, q E subK of different atoms, there existm E N(p) and n E N(q) such that m A n = OK. Let R E RK be a rasterwith R - r and R - s where r, s E subK are atoms. Then N(r) C Rand N(s) C R, hence m A n > oK whenever m E N(r) and n E N(s).Therefore, r = s. We have shown that K is raster separated.Conversely, let c be additive and let subK be a Boolean algebra. Supposethere is a pair p, q E subK of different atoms such that m A n > oK whenever
m E N(p) and n E N(q). Put B = N(p) U N(q). Then B C subK iscentered because N(p) and N(q) are filters (by Proposition 2.2(9)). Sincewe have both [B] -> p and [B] -> q, K is not raster separated. 0
Theorem 4.4 Let K be a )C-object such that subK is a Boolean algebra.If r = A{cK(n) ; n E N(r)} for each atom r E subK, then K is rasterseparated, and vice versa provided that c is additive and subK is atomic.
Proof. Let r = A{cK(n) ; n E N(r)} for each atom r E subK. Let
p, q E subK be different atoms. Then
Hence, there is n E N(P) wio q f cK(n). Thus, q cK(n) and we have
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n E N(q). As n A n = OK, K is raster separated by Theorem 4.3.Conversely, let c be additive, let subK be atomic, and let K be raster sepa-rated. Given an atom r E subK, we have r n{cK(n) ; n E N(r)}. Admitthat r A{CK(n) ; n E N(r)}. Then there is an atom p E subK, p differentfrom r, such that p A{cK(n) ; n E N(r)}. Thus, by Theorem 4.3, thereare m E N(p) and n E N(r) with m A n = OK. Consequently, n> m.This yields n E N(p), i.e., p cK(n). Therefore, p f cK(n), which is acontradiction. 0
Now, we will proceed to the study of raster compactness. Definition4.1 (b) and Proposition 3.6 immediately result in
Proposition 4.5 Let K be a IC-object such that subK is a Boolean algebra.If for each R E RK there exist S E RKwith8 2 R and an atom rn E subKsuch that S-> m, then K is raster compact, and vice versa provided thatsubK is atomic, c is additive and RK is replaced by the conglomerate of allfilters on subK.
Corollary 4.6 Let c be additive and K be a IC-object such that subK isan atomic Boolean algebra. Then K is raster compact if and only if eachultrafilter R E RK converges to an atom of subK.
Theorem 4.7 Let K be a IC-object. If K is raster compact, then 1B T > OK
for each centered class T g subK of c-closed subobjects of K, and viceversa provided that c is idempotent.
Proof. Let K be raster compact. Admit that there exists a centered classT C subK of c-closed subobjects of K such that AT = OK. Then [T] ERK and A {CK (P) ; p E [T]l :5 A{cK(P) ; p E T}= 1B r =OK. Hence theonly clustering of [T] is OK, which is a contradiction.Conversely, let c be idempotent and let A T >oK for each centered classT g subK of c-closed subobjects of K. Let R E RK and let S C ’R be theclass of all c-closed elements of R. Then S is centered, hence A S > oK.
Since A S=A{cK(r) ; r E R}, we have A{cK(r) ; r E R} > OK. As
A{cK(r) ; r E RI is clearly a clustering of R, R is raster compact. 0
Theorem 4.8 Let c be idempotent and hereditary, and let m : M -> K be ac-closed embedding in IC. If K is raster compact, then M is raster compacttoo.
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Proof. Let K be raster compact and T C subM be a centered class of c-closed subobjects of M. Let t E T be an arbitrary element. Then CM(t) =m-1(cK(m o t)) because c is hereditary. It follows that m-1(cK(m o t)) = tas cM(t) = t. Thus, since cK(m) = m, we have (cx(m))-1(cK(m 0 t)) = t.But we also have cK(mot) cK(m) because mot m. Hence cK(mot)=cK(m) ot = mot. Therefore, mot is c-closed. Further, for any tl, t2, ..., tk ET (k EN) we have m o t 1 A m o t2 A...A m o tk> m(t1 A t2 A...A tk) > oK.
Consequently, m o T E subK is centered. It follows that A (m 0 T)> oK.Thus, as A(m o T) = m o A T, there holds A T> om. By Theorem 4.7,M is raster compact. 0
In the proof of Theorem 4.8, the hereditariness of c has been used only forshowing that a composition of a pair of c-closed subobjects is c-closed. Asc is supposed to be idempotent, Theorem 4.8 remains valid when replacingthe hereditariness of c with the so-called weak hereditariness of c (see [14],2.4).
Theorem 4.9 Let .6 be stable under pullbacks, c be idempotent and f : K -L be a )C-morphism. If K is raster compact and f E e, then L is rastercompact too.
Proof. Let T c subL be a centered class of c-closed subobjects of L.Then f-1 (T) is a centered class of c-closed subobjects of K and there-fore, Since we have
A T = f(f-1(A T)) > OLe Now, the statement follows from Theorem 4.7.0
Theorem 4.10 Let c be additive, idempotent and hereditary. Let m : M->K be an embedding in JC where M is raster compact, K is raster separatedand subK is an atomic Boolean algebra. Then m is c-closed.
Proof. If m = OK, then the statement is trivial. Let rra > 0 K and let
p cK(m) be an atom. By Proposition 2.5, n A rn > OK for each n EAs c is hereditary,
for each n E N(p) there holds m-1(cK(m o m-1(n))) = cM(m-1(n)).Thus, each element of T is a c-closed subobject of M and for any n E
Consequently, given
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because m A nl A
n2 A ... A nk > OK (as nl A n2 A ... A nk E N(p) by Proposition 2.2(9)). Itfollows that T is centered. Since m o m-1(n) = m(m-1(n)) n, we haveA T A{m-1(cK(n)) ; n E N (P)} = m-i(A{cK(n) ; n E A (p)}). Thus,by Theorems 4.4 and 4.7, OM A T m-l(p). As p A rri = rrz o m-1 (p),we get p A m > oK. Hence p m because p is an atom. Therefore,cK(m) m.~
Corollary 4.11 Let 9 be stable under pullbacks and IC have embeddingsand (E, Emb M) -factorization structure. Let c be additive, idempotent andhereditary and f : K -> L be a IC-morphism where K is raster compactand L is raster separated with the property that subL is an atomic Booleanalgebra. Then f is c-preserving.
Proof. Let m E subK. Then there is a IC-object M such that cK(m) :M -3 K is an embedding in 1C. As cK(m) is c-closed, M is raster com-pact by Theorem 4.8. Further, there is a 1C-object N such that f (cK(m)) :N -3 L is an embedding in 1C. Let e : |M|-> |N| be the 6-part ofthe (E, M) -factorization of f o cK(m). By the assumptions, e is a 1C-
morphism. Thus, N is raster compact by Theorem 4.9. According to The-orem 4.10, f (cK(m)) is c-closed. From f(m) f (cK(m)) it follows thatcL(f(m)) cL(f (cK(m))) = f (cK(m)). As the converse inequality isclearly fulfilled, f is c-preserving. 0
Remark 4.12 Let the assumptions of Corollary 4.11 be satisfied and let IChave the property that each IC-morphism which is a c-preserving X-isomor-phism is a 1C-isomorphism. Then f is a 1C-isomorphism whenever it is anX-isomorphism. Moreover, let |K|= I L and suppose that cK CL impliesthat id|K| is a K-morphism id|K| : K - L. Then cK CL implies cK = CLby Corollary 4.11 (putting f = id|K|). Thus, given an X-object X, in theclass of all cK with K a raster separated 1C-object such that IKI = X, cKwith K raster compact are maximal (provided that the class is nonempty).
As an immediate consequence of Theorem 4.9 we get
Corollary 4.13 Let .6 be stable under pullbacks and c be idempotent. LetK = IIiEI Ki be a product in K such that all projections pri : K -> Ki, i E
I, belong to E. If K is raster compact, then Ki is raster compact for eachi E I.
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The following statement is a converse of Corollary 4.13.
Theorem 4.14 Let in X the non-trivial objects be stable under products,let c be additive and K = flic :, Ki be a product in IC such that subK is aBoolean algebra and subKi is an atomic Boolean algebra for each i E I.Let each ultrafilter on subK be convergence-compatible with K. If Ki israster compact for each i E I, then K is raster compact too.
Proof. Let Ki be raster compact for each i E I and let R E RK be anultrafilter. Then [pri(R)] is an ultrafilter for each i E I. As subK is atomic,subKi is atomic too for each i E I by Lemma 1.2. By Corollary 4.6, for eachi E I there is an atom mi E subKi such that [pri (R)] -> mi. Since in X thenon-trivial objects are stable under products, we have IIiEI mi > OK- Thus,there is an atom m fliEI mi. Then pri(m) :5 mi and thus [pri(R)] -pri(m) for each i E I. As R is convergence-compatible with K, we haveR-> m. Hence, K is raster compact by Corollary 4.6. D
Remark 4.15 If JC = Top, the assumptions of all statements of this sectionare fulfilled and these statements give well-known results. The property pre-sented in Theorem 4.10 is sometimes referred to as the absolute closednessof M, and that of Remark 4.12 as minimality of compact spaces (note thattopologies on a given set are usually ordered conversely to the correspondingKuratowski closure operators). Notice also that Theorem 4.14 is Tichonoff ’stheorem transposed to our settings.
5 c-separation and c-compactnessOur investigation of raster separation and raster compactness with respectto c would be rather incomplete without discussing their relationships to c-separation and c-compactness, respectively. The aim of this section is tostudy these relations.
Definition 5.1 [10] A K-object K is called
(a) c-separated if the diagonal morphism 6K : IKI -t IKI x IKI is c-closed,
(b) c-compact if the projection prL : K x L -> L is c-preserving for everyK-object L.
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Remark 5.2 It is well known [10] that a 1C-object K ic c-separated if andonly if, for each pair f, g : K - L of 1C-morphisms and each m E subK,f o m = g o m implies f 0 CL(M) = go CL(M)-
Theorem 5.3 Let c be additive and K be a IC-object such that sub(K x K) isan atomic Boolean algebra and both the projections pri : |K|x |K|->|K|,i = 1, 2, fulfill pri o rra E M whenever m E sub(K x K) is an atom. If K israster separated, then it is c-separated.
Proof Let K be raster separated. If CKxK(6K) = OK, then the statement istrivial. Let cKxK(dK) > oK and let m E sub(K x K) be an atom with m cKxK (dK). Then, by Lemma 3.9, there is a filter base B C sub(K x K) with[B] -> m such that q :5 JK for each q E B. Consequently, for each q E Bthere exists tq E M such that q = 6K o tq. We have B = 16K o tq; q E B},hence pri(B) = {pri(dK o tq) ; q E B} for i = 1, 2. From prl o 6K o tq =pr2 o 6K o tq it follows that prl(bK o tq) = pr2(bK o tq) for each q E B.This yields [pri(B)] = [pr2(B)]. By Theorem 3.10, (pri(B)] -> pri(m) fori = l, 2. Thus, since pri(m) = prz om, i = 1,2, are atoms of sub (I K x I K 1)by Lemma 1.2, we have prl o m = pr2 o m. This results in m 6K because6K is an equalizer of prl and pr2. Therefore, cKxK(dx) 6K, i.e., 6K isc-closed. Hence K is c-separated. 0
Theorem 5.4 Let K be a )C-object such that all atoms of subK have thesame domain (up to isomorphisms), sub(K x K) is a Boolean algebra, andeach raster on K x K is convergence compatible with K x K. If K isc-separated, then it is raster separated.
Proof Let K be c-separated, R E RK, R-> m and R-> p where m, p Esub(K x K) are atoms. Put S = [dK (R)] and let prl, pr2 : |K| x IKI -> IKI be the two projections. Then S E RKXx and [pri(S)]- R for i = 1, 2.So, we have (prl(S)] -> m and [pr2(S)] -> p. Putting q = (m, p) we getan atom q E sub(K x K) by Lemma 1.3. As m = pri o q = prl (q) andp = pr2 o q = pr2(q), the convergence compatibility of S results in S -> q.Let s E S. Then there exists r E R with s> 6K(r). Thus, we haves A dK > 6K(r) A 6K = 6K(r) > oK because r > oK. By Proposition 3.4,q cKxK(dK) = 6K. Consequently, there exists t E .M with q = 6K o t.We have m = pri o q = prl o dK o t = Pr2O6KOt = pr2 o q = p. Therefore,K is raster separated. 0
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Theorem 5.5 Let c be additive and subL be an atomic Boolean algebra foreach )C-object L. Let K be a IC-object satisfying the following condition:
Given a JC object L, an atom y E subL and a subobject m E sub(K xL) with prL(cKxL(m)) A y = OL, for each atom x E subK thereare subobjects ux E subK and Vx E subL, Ux c-closed, such thatux A x = OK, cL(vx) A y = OL, and cKxL(m) pr"Kl(ux) V pri-1L(vc).
If K is raster compact, then it is c-compact.
Proof. Let K be raster compact, L be a JC-object and m E sub(K x L). IfprL(cKxL(m)) = idL, then we clearly have cL(prL(m)) prL(cKXL(m)).Let prL(cKXL(m)) idL. Then prL(cKxL(m)) > OL. Let y E subLbe an atom with y prL(cKxL(m)), i.e., with prL(cKxL(m)) A y = OL-For each atom x E subK, let ux E subK and vx E subL be the sub-
objects from the condition of the statement. Then A {ux ; x E subK isan atom} = OK (because otherwise there is an atom xo E subK withx0 Ux for each atom x E subK, which is a contradiction with uxo A xo =
OK). Thus, by Theorem 4.7, there is a finite set {x1, ..., ck} of atoms ofsubK such that Aki=1 uxi = oK. Put V = vk VXi. Then cL(v) A y =
cL (Vki=1 vxJ A y = Vki=1 (CL (vxi) A y) = oL. Consequently, v E N(y). Fur-ther, we have c
hence prL(cKxL(m)) v.
This yields v prL(cKxL(m)), i.e., v A prL(CKxL(m)) = OLe It follows thatv A prL(m) = OL. By Proposition 2.5, y A cL(prL (m)) = OLe Consequently,y E cL(prL(m)). We have shown that cL(prL(m)) > prL(cKxL(m)). There-fore, cL(prL(m)) prL(cKxL(m)) and the proof is complete. 0
Theorem 5.6 Let c be idempotent and K be a IC-object with the propertiesthat subK is a Boolean algebra and for any centered subclass F C subK ofc-closed subobjects of K there exist a IC-object L and a c-dense subobjectm : IKI --t ILl of L such that the following conditions are satisfied:
(1) sub(K x L) is atomic.
(2) For any atom z E sub(K x L), from p E N(prK(z)) and q EN(prL(z)) itfollows that p x q E N(z).
(3) There exists a subobject y E subL with y > OL, y A m = OL, and
y V m (s) E N(y) for each s E F.
297
If K is c-compact, then it is raster compact.
Proof. Let K be c-compact and F C subK be a centered class of c-closedsubobjects of K. Put d = (idK, m). Then m = prL(d) prL(cKxL(d)). Asm is dense, we have y cL(m). Consequently, y cL(prL(cKxL(d)))=prL(ckxL(d)) because prL : K x L - L is c-preserving and c is idempotent.Thus, by Lemma 1.1, cKxL(d) A pr-1L(y)> OK xL. Let z E sub(K x L) be anatom with z cKxL(d) A pr-1L(y). Then z cxxL(d) and prL(z) y. Puta = prK(z) and qs = y V m(s) for each s E F. By Lemma 1.2, a E subKis an atom. Let p E N(a). Since qs E N(y), we have p x qs E N(z) foreach s E 7. By Proposition 2.5, w A d > oKxL for each w E N(z). Thus,(p x (y V m(s))) A d > oKxL for each s E 7. Hence, there is an atomvs E sub(K x L) with us (p x (y V m(s))) A d for each s E 7. Asus d, there is an element us E subK, us > oK, with vs = d o Us. We have
prKous = prKo(idK, m)ous =us andprLovs = prLO(idK, m)ous = mous.From us P x (y V m(s)) it follows that us prK(p x (y V m(s))) andm(us) prL(P x (y V m(s))) (for each s E r). Now, using the (E, M)-diagonalization property, we get us p and m(us) y V m(s), i.e., us
each s E :F. Therefore, by Proposition 2.5, a cK(s) = s for each s E 7.Hence AF>oK and by Theorem 4.7, K is raster compact. 0
Example 5.7 If )C = Top, the assumptions of each of the Theorems 5.3-5.6 are satisfied and the Theorems then give well-known results. Theorems5.3 and 5.5 are also valid for example for the larger category IC = PrTopof pretopological spaces in the sense of Cech (i.e., closure spaces from [6]).The assumptions of Theorem 5.5 are satisfied whenever IC is a full topo-logical subcategory of Top (the subobjects Ux and ux are then obtained ascomplements of certain open neighborhoods of x and y, respectively - see[15]). As for Theorem 5.6, its assumptions are satisfied, for example, when-ever 1C is the category of Ti-spaces or the category of normal spaces (thetopological space L is then defined to be the space with ILI = |K| U {y}where y E |K| is a point and the open sets in L are just the open sets inK and the sets of the form {y} U T U X where T is a finite intersection ofelements of 7 and X C |K| is a subset - see [ 15] again). On the other hand,there hardly exist topological categories which are not subcategories of Topand fulfill the conditions of Theorem 5.5 or 5.6.
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Remark 5.8 a) The assumptions of Theorems 5.3 and 5.4 are quite natural(especially if K is a construct), thus there is a strong relationship betweenraster separation and c-separation. But this is not true for raster compactnessand c-compactness in general (if c is not a Kuratowski closure operator).Theorems 5.5 and 5.6 are presented here mainly for the sake of complete-ness, and are by no means our major concern in this paper.b) The results of section 4 show that raster compactness behaves more de-cently than c-compactness because the former preserves all basic proper-ties of the usual topological compactness transposed to our setting. This isa consequence of the fact that raster compactness is closer to the classical
Lebesgue definition of compactness than c-compactness which is based onthe Kuratowski-Mrowka characterization. But, in contrast to c-compactness,raster compactness can be considered only for such a category k with a clo-sure operator which (together with its underlying category X) fulfills all theconditions assumed in the first section
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ERALDO GIULI, DIPARTIMENTO DI MATEMATICA, UNIVERSITA DIL’ AQUILA, 67100 L’ AQUILA, ITALY
e-macil address : giuli @ univaq.itJOSEF SLAPAL, DEPARTMENT OF MATHEMATICS, BRNO UNIVER-
SITY OF TECHNOLOGY, 616 69 BRNO, CZECH REPUBLIC